| | Quadratic | Linear with kernel weight |
| (Intercept) | 0.769*** | 0.819*** |
| (0.034) | (0.015) |
| Income_Centered | -11.567 | -23.697*** |
| (8.101) | (3.219) |
| Participation | 0.093** | 0.033 |
| (0.044) | (0.021) |
| I(Income_Centered^2) | 562.247 | |
| (401.982) | |
| Income_Centered × Participation | 19.300* | 26.594*** |
| (10.322) | (4.433) |
| Participation × I(Income_Centered^2) | -101.103 | |
| (502.789) | |
| * p < 0.1, ** p < 0.05, *** p < 0.01 |
]
---
# Fuzzy regression discontinuity
<br>
.vcenter[
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_16c3611432ee1_files/figure-html/unnamed-chunk-10-1.png" alt="8: A causal diagram that fuzzy regression discontinuity works for" width="60%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">8: A causal diagram that fuzzy regression discontinuity works for</p>
</div>
]
---
# Fuzzy regression discontinuity
<br>
**Sharp RDD vs. FRDD**
<br>
- In FRDD, the data cannot simply be limited to the area around the cutoff to control for `\(\text{Running Variable}\)`
→ Doing that would lead us to understate the effect!
- Instead we apply IV:
- The first stage uses `\(\text{AboveCutoff}\)` as an instrument for `\(\text{Treated}\)`
- Estimate regression discontinuity equations as for the sharp RDD in the second-stage equation
---
# Fuzzy regression discontinuity
<br>
**IV estimation of FRDD**
<br>
IV divides the effect of the instrument on the outcome by the effect of the instrument on the endogenous/treatment variable.
→ The effect of being *above* the cutoff on the outcome is scaled but divided to account for the fact that being above the cutoff only leads to a partial increase in treatment rates.
---
# Fuzzy regression discontinuity
.vcenter[
.blockquote[
### Example: Effect of mortgage subsidies on home ownership (Fetter 2013).fn[3]
- Fetter’s main research question is how much of the increase in the home ownership rate in the mid-century US was due to mortgage subsidies given out by the government
- He considers people who were about the right age to be veterans of major wars like WWII or the Korean war:
Anyone who was a veteran of these wars received special mortgage subsidies.
]]
.footnote[[3] Fetter, Daniel K. 2013. *How Do Mortgage Subsidies Affect Home Ownership? Evidence from the Mid-Century GI Bills*. American Economic Journal: Economic Policy 5 (2): 111–47.]
---
# Fuzzy regression discontinuity
.vcenter[
.blockquote[
### Example: Effect of mortgage subsidies on home ownership (Fetter 2013)
- There is an age requirement to join the military:
If one is born one year too late to join the military to fight in the Korean war, then he will not get these mortgage subsidies (or at least far fewer veterans were eligible).
→ Discontinuity based on birth year.
- The “treatment” of being eligible for mortgage subsidies would only apply to some people born at the right time:
Treatment rates jump from 0% to some value below 100% (fuzzy).
- Veteran status at this margin increases home ownership rates by 17%
]]
---
# Fuzzy regression discontinuity
</br>
.blockquote[
### Example: Effect of mortgage subsidies on home ownership (Fetter 2013)
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_16c3611432ee1_files/figure-html/unnamed-chunk-11-1.png" alt="9: Eligibility for mortgage subsidies for being a Korean war veteran and home ownership from Fetter (2013)" width="70%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">9: Eligibility for mortgage subsidies for being a Korean war veteran and home ownership from Fetter (2013)</p>
</div>
</br>
]
---
# RDD
<br>
**Placebo tests**
<br>
- The astonishing thing about regression discontinuity is that it closes all back doors, even the ones that go through variables which cannot be measured
- That is the whole idea:
Isolate variation in such a narrow window of the running variable so that it is plausible to claim that the *only* thing changing at the cutoff is treatment—and by extension anything that treatment affects (like the outcome)!
---
# Placebo tests
</br>
**Idea**
Anything we would normally use as a control variable should not affect treatment.
<br>
**Procedure**
- Run the regression discontinuity model on plausible control variables
- If an effect is found, the original RDD might not have been right. This might indicate that our assumption about randomness at the cutoff is violated.
---
# Placebo tests
</br>
**Procedure**
<br>
Keep in mind that, since we can run placebo tests on a long list of potential placebo outcomes, it is likely that we find a few nonzero effects just by random chance.
→ If one test a long list of variables and find a few differences, that is not a fatal problem with the design.
In these cases, it is better to add the variables with the failed placebo tests to the model as control variables.
---
# Placebo tests
.vcenter[
.blockquote[
### Example: Manacorda et al. (2011)
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_16c3611432ee1_files/figure-html/unnamed-chunk-12-1.png" alt="10: Performing regression discontinuity with controls as outcomes for a placebo test" width="70%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">10: Performing regression discontinuity with controls as outcomes for a placebo test</p>
</div>
]]
---
# The density discontinuity test
</br>
**Random assignment may fail**
</br>
There are two ways manipulation could happen:
- First, whoever (or whatever) is in charge of setting the cutoff value might do so with the knowledge of exactly who it will lead to getting treated.
- Second, individuals themselves likely have some control over their running variable. Sometimes they have *direct control* and sometimes they have *indirect control*.
---
# The density discontinuity test
</br>
**Random assignment may fail**
</br>
In the case of indirect control we do have a test we can perform to check whether manipulation seems to be occurring at the cutoff.
For this we inspect the distribution of the running variable around the cutoff:
- If the running variable was randomly assigned without regard for the cutoff we expect its distribution to be smooth
- A distribution that seems to have a dip just to one side of the cutoff, with those observations sitting just on the other side, this may indicate manipulation
---
# The density discontinuity test
</br>
**Steps**
</br>
- Estimate the density of the treatment variable. Allow that density to have a discontinuity at the cutoff.
- Look for a significant discontinuity at the cutoff
- Do graphical inspection of the density
<br>
**A big discontinuity is not a good sign. It implies manipulation.**
---
# The density discontinuity test
.vcenter[
.blockquote[
### Example: Manacorda et al. (2011)
</br>
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_16c3611432ee1_files/figure-html/unnamed-chunk-13-1.png" alt="11: Distribution of the binned running variable" width="70%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">11: Distribution of the binned running variable</p>
</div>
<br>
]]
---
# Regression kink
<br>
- Another thing to inspect is a change in the slope of the relationship between the outcome and the running variable.
- The treatment administered at the cutoff does not make the outcome itself change/jump—it changes the *strength* of the relationship between outcome and the running variable
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_16c3611432ee1_files/figure-html/unnamed-chunk-14-1.png" alt="12: Regression kink on simulated data" width="55%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">12: Regression kink on simulated data</p>
</div>
---
# Regression kink
.vcenter[
.blockquote[
### Example: Effect of unemployment benifits on job findings (Card et al. 2015.fn[4])
- Card et al. use data from Austria, where unemployment insurance benefits are 55% of regular earnings but there is an upper limit of benefits.
- So regular earnings (running variable) positively affect the amount of unemployment insurance received (treatment), up to the cutoff, at which point the effect of regular earnings on your unemployment insurance payment becomes zero.
- If generous unemployment benefits make people take longer to find a new job, we would expect to find a positive relationship between regular earnings (running variable) and time-to-find-a-new-job (outcome) up to the point of the cutoff, and then it should be flat after the cutoff.
Card et al. find evidence of such an effect.
]]
.footnote[[4] Card, David, David S Lee, Zhuan Pei, and Andrea Weber. 2015. *Inference on Causal Effects in a Generalized Regression Kink Design*. Econometrica 83 (6): 2453–83.]
---
# Regression kink
.vcenter[
.blockquote[
###Example: Treatment-has-a-kink regression kink design (Bana et al. 2020.fn[5])
- Bana et al. look at the impact of paid family leave (Elternzeit) in California
- In California, the state pays 55% of regular earnings up to a maximum benefit amount. Family leave payment (treatment) increases with regular earnings (running variable), until a maximum amount (cutoff) is reached
- After that, additional regular earnings does not increase the family leave payment
]]
.footnote[[5] Bana, Sarah H, Kelly Bedard, and Maya Rossin-Slater. 2020. *The Impacts of Paid Family Leave Benefits: Regression Kink Evidence from California Administrative Data*. Journal of Policy Analysis and Management 39 (4): 888–929.]
---
# Regression kink
.vcenter[
.blockquote[
### Example: Treatment-has-a-kink
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_16c3611432ee1_files/figure-html/unnamed-chunk-15-1.png" alt="13: Paid family leave benefits and pre-leave Earnings" width="60%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">13: Paid family leave benefits and pre-leave Earnings</p>
</div>
<br>
]]
---
# Regression kink
.vcenter[
.blockquote[
### Example: Treatment-has-a-kink
- Treatment on the y-axis can be replaced with some outcome to see whether it also changes slope. If so, that is evidence of an effect.
- Bana et al. look at a various outcome variables, e.g.,
- how long the mothers stay on family leave.
- whether they use family leave again in the next three years, conditional on going back to work in the meantime.
]]
---
# Regression kink
.vcenter[
.blockquote[
### Example: Treatment-has-a-kink regression kink design
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/RDD/renderthis_16c3611432ee1_files/figure-html/unnamed-chunk-16-1.png" alt="14: Log leave duration or proportion claiming leave again and pre-leave earnings" width="80%" style="display:block; margin-right:auto; margin-left:auto;" />
<p class="caption">14: Log leave duration or proportion claiming leave again and pre-leave earnings</p>
</div>
<br>
]]
---
# When running variables misbehave
<br>
**Granularity and heaping**
<br>
Two potential issues are
- ... the running variable being too **granular:** it is measured at a too coarse level.
- ... the running variable exhibiting **heaping:** it has some values that are suspiciously much more common than others.
---
# When running variables misbehave
<br>
**Granularity**
<br>
- **Example**
on measuring annual income, it can be said that a person earned $40,231.36 last year, or can also be said say that he earned $40,231, or say that he earned $40,000. Or we could even say that he earned '$40-50,000'. Or, say 'less than $100,000'. These are measurements of his income in decreasing order of granularity.
- **Solution**
The real solution here is 'find a running variable that is granular enough'. But no variable is infinitely granular. So when worried about granularity, it is better to pick an estimator that will account for that granularity.
---
# When running variables misbehave
<br>
**Heaping**
<br>
**Non-random heaping** is when the running variable seems to be much more likely to take certain values than others. Often this can come in the form of rounding.
- **Example**
If one asks people how old they are and the vast majority of them are going to give a round number, like '36 years old'. However, some of them will be more precise and say '36 years, eight months, and two days'.
- **Solution**
A common approach to this is **donut hole regression discontinuity**, where one simply drops observations just around the cutoff so as to clear out heaps near the cutoff.
---
# Dealing with bandwidths
<br>
How far away from the cutoff can we get and still have comparable observations on either side of it?
The answer to this question is a tradeoff:
- Picking a bandwidth around the cutoff that is too wide we bring in observations that are not comparable making the estimates less believable and more biased
- Picking a bandwidth that is too narrow we will end up estimating the effect on hardly any data, resulting in a noisy estimate
- This is a **efficiency-robustness tradeoff**
---
# Dealing with bandwidths
<br>
**How to proceed?**
<br>
- **Just pick a bandwidth**
This is a very common approach. Although perhaps it is becoming less common over time.
- **Pick a bunch of bandwidths**
This sensitivity test-based approach is also fairly common. For this approach, the biggest bandwidth is picked which seems to makes some sense for the research design, and then start shrinking it. Each time, estimate the regression discontinuity model.
- **Data based bandwidth selection**
Pick an objective of 'what a good bandwidth looks like' and then use the data to figure out how wide a bandwidth is the best one by that criterion
---
# Dealing with bandwidths
<br>
**Data-based bandwidth selection**
<br>
- **Cross-validation**
The basic idea is to try to get the best out-of-sample predictive power. So the data is splitted into random chunks. For each chunk, the model is estimated using all the other chunks, and then make predictions for the now out-of-sample chunk which is left out. Repeat this for every chunk, and then see overall how well your out-of-sample prediction went. Then, repeat that whole process for each potential bandwidth and see which one does the best.
- **Optimal bandwidth rules**
This approach takes as its goal getting the best prediction right at the cutoff, just on either side.